**Matrix Multiplication Using Block Matrices**We have two matrices, $ A $ and $ B $, expressed as block matrices:\[ A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 4 & 5 \\ 0 & 1 & -5 & 5 & -4 \\ 0 & 0 & -1 & -5 & -6 \end{bmatrix} \]\[ B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} 6 & -2 & -2 & -5 \\ 4 & 2 & 5 & -5 \\ 6 & 3 & 6 & 5 \\ 4 & -1 & 5 & 1 \\ 1 & 1 & -4 & 0 \end{bmatrix} \]**Objective:**Let $ C = AB $. Then $ C $ can be computed by block multiplication:\[ C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{bmatrix} \]**Instructions for Calculation:**1. **Compute the Block Products:** - $ A_{11}B_{11} $ (top left matrix block) - $ A_{12}B_{21} $ (top left supplementary matrix block)2. **Use these products to find $ C_{11} $, and then evaluate the remaining blocks of $ C $ similarly:**\[ C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} \]**

Answered: Image /qna-images/answer/9311a1b0-21eb-4bc6-9697-6c021a08719a.jpg

Compute the following block products: A11 B11 || 1 0 0 --**-**4] 0 1 -5 5 0 0 -1 -5 -6 Let C = AB. Then C can be computed by block multiplication: c = [C₁1 C1₂] - C = [ C22 1 c=[CH CH₂2]- C21 A = [ALI B = A11 A12 A12B21 B11 B12 B21 B22 -1 = Use them to find C₁1, and then evaluate the remaining blocks of C similarly: 6 -2 -2 4 2 6 3 4 −1 1 1 [A11B11+A12B21 A21B11 + A22B21 -5 5 -5 non 6 5 5 1 -4 0 5 A11B12 + A12B22 A21 B12 + A22 B22

Compute the following block products: A11 B11 || 1 0 0 ---4] 0 1 -5 5 0 0 -1 -5 -6 Let C = AB. Then C can be computed by block multiplication: c = [C₁1 C1₂] - C = [ C22 1 c=[CH CH₂2]- C21 A = [ALI B = A11 A12 A12B21 B11 B12 B21 B22 -1 = Use them to find C₁1, and then evaluate the remaining blocks of C similarly: 6 -2 -2 4 2 6 3 4 −1 1 1 [A11B11+A12B21 A21B11 + A22B21 -5 5 -5 non 6 5 5 1 -4 0 5 A11B12 + A12B22 A21 B12 + A22 B22

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Matrix Multiplication Using Block Matrices**

We have two matrices, $ A $ and $ B $, expressed as block matrices:

\[
A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 4 & 5 \\ 0 & 1 & -5 & 5 & -4 \\ 0 & 0 & -1 & -5 & -6 \end{bmatrix}
\]

\[
B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} 6 & -2 & -2 & -5 \\ 4 & 2 & 5 & -5 \\ 6 & 3 & 6 & 5 \\ 4 & -1 & 5 & 1 \\ 1 & 1 & -4 & 0 \end{bmatrix}
\]

**Objective:**
Let $ C = AB $. Then $ C $ can be computed by block multiplication:

\[
C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{bmatrix}
\]

**Instructions for Calculation:**

1. **Compute the Block Products:**
- $ A_{11}B_{11} $ (top left matrix block)
- $ A_{12}B_{21} $ (top left supplementary matrix block)

2. **Use these products to find $ C_{11} $, and then evaluate the remaining blocks of $ C $ similarly:**

\[
C = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix}
\]

**

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Step 1: Determination of all blocks

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Step 3: Finding $c_12$

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Step 4: Finding $c_21$

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Step 5: Finding $c_22$

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